ASTRONOMICAL FORMULAE


TELESCOPES

APERTURE
D = F/f

where d is the aperture of the objective
F is the focal length of the objective
f is the f-number (f/) of the objective


MAGNIFICATION: BY FIELDS
M = Alpha/Theta

where M is the magnification
Alpha is the apparent field
Theta is the true field

Apparent Field: the closest separation eye can see is 4', more
practically 8-25', 1-2' for good eyes. The Zeta Ursae Majoris
double (Mizar/Alcor) is 11.75'; Epsilon Lyrae is 3'.

True Field (in °) = 0.25 * time * cos of the declination
(in ') = 15 * time * cos of the declination
where time is the time to cross the ocular
field in minutes
A star therefore moves westward at the following rates:
15° /h (1.25°/5 min) at 0° declination
13° /h (1.08°/5 min) at 30° declination
7.5°/h (0.63°/5 min) at 60° declination.


MAGNIFICATION: BY FOCAL LENGTHS
M = F/f

where M is the magnification
F is the focal length of the objective
f is the focal length of the ocular

At prime focus (ground glass), magnification is 1x for
each 25 mm of F

MAGNIFICATION: BY DIAMETER AND EXIT PUPIL
M = D/d

where M is the magnification
D is the diameter of the objective
d is the exit pupil (5-6 mm is best; 7 mm may not
produce a sharp outer image)

The scotopic (dark-adapted) aperture of the human pupil is
typically 6 (theoretically 7, 5 if over age 50) mm. Since the
human pupil has a focal length of 17 mm, it is f/2.4 and yields
0.17 per mm of aperture. 2.5 mm is the photopic (light-adapted)
diameter of the eye.


EXIT PUPIL
d = f/f-number

(by substituting F/f for M)
where d is the exit pupil
f is the focal length of the ocular
f-number is the f-number (f/) of the objective

By substituting d=7 (the scotopic aperture of the human pupil)
and multiplying it by the f-number, the longest useful focal
length of the ocular is given.

LOW-POWER LAW FOR LIMITING MAGNIFICATION
M = D/6 = 17*D

(by substituting 6 mm for d and taking the reciprocal)
where M is the minimum magnification without wasting light
for a dark-adapted eye (17x per mm of aperture)
D is the diameter of the objective in mm


HIGH-POWER LAW FOR LIMITING MAGNIFICATION
M = D/0.63 = 158*D

(by substituting 0.63 mm, the minimum diameter to which the
average pupil can contract, for d and taking the reciprocal)

where M is the maximum theoretical magnification (158x per mm
of aperture); the maximum practical magnification is +50%).

LIMITING VISUAL MAGNITUDE (LIGHT-GATHERING POWER)
m = 6.5-5 log Delta+5 log D
= 2.7+5 log D (assuming transparent dark-sky conditions and
magnification >= 1D in mm)

where m is the approximate limiting visual magnitude
Delta is the pupillary diameter in mm (accepted as 7.5)
D is the diameter of the objective in mm

RELATIVE LIGHT EFFICIENCY (TWILIGHT FACTOR)
Relative Brightness Value = d^2 = (D/M)^2

where the larger the relative brightness value, the better
the instrument (e.g., binoculars) is for viewing in twilight
or for astronomical use at dusk (low light conditions only)
d is the diameter of the exit pupil
D is the diameter of the objective
M is the magnification


ANGULAR RADIUS OF AIRY (DIFFRACTION) DISC
r = (1.12*Lambda*206265)/D
= 127.1/D

(the second formula is based on Lambda = 0.00055 for yellow)
where r is the angular radius (one-half the angular diameter)
of the Airy disc (irreducible min. size of a star disc in ")
Lambda is the wavelength of the light in mm
206265 is the number of " in a radian
D is the diameter of the objective in mm

LINEAR RADIUS OF AIRY (DIFFRACTION) DISC
r = 0.043*Lambda*f

where r is the linear radius (one-half the linear diameter)
of the Airy disc in mm
Lambda is the wavelength of light in mm (yellow 0.00055)
f is the f-number (f/) of the objective


DAWES LIMIT (SMALLEST RESOLVABLE ANGLE, RESOLVING POWER)
Theta = 115.8/D

where Theta is the smallest resolvable angle in "
D is the diameter of the objective in mm

Atmospheric conditions seldom permit Theta < 0.5". The Dawes
Limit is one- half the angular diameter of the Airy (diffraction)
disc, so that the edge of one disc does not extend beyond the
center of the other). The working value is two times the Dawes
Limit (diameter of the Airy disc), so that the edges of the two
stars are just touching.


MAGNIFICATION NEEDED TO SPLIT A DOUBLE STAR
M = 480/d

where M is the magnification required
480 is number of seconds of arc for an apparent field of
8 minutes of arc
d is the angular separation of the double star

About the closest star separation that the eye can distinguish
is 4 minutes of arc (240 seconds of arc). Twice this distance, or
an 8-minute (480- second) apparent field angle, is a more
practical value for comfortable viewing. In cases where the comes
is more than five magnitudes fainter than the primary, you will
need a wider separation: 20 or 25 minutes of arc, nearly the
width of the moon seen with the naked eye.


RESOLUTION OF LUNAR FEATURES
Resolution = (2*Dawes Limit*3476)/1800)
Dawes Limit * 38.8

where Resolution is the smallest resolvable lunar feature in km
2*Dawes Limit is the Airy disc (a more practical working
value is twice this)
1800 is the angular size of the moon in "
3476 is the diameter of the moon in km


LIGHT GRASP
Light Grasp = (D/d)^2*Pi
= 7*D^2

where Light Grasp is times that received by the retina
D is the diameter of the objective in mm
d is the diameter of the eye's pupillary aperture in mm
(accepted value 7.5)
pi is the transmission factor (approximately equal to
62.5% for the average telescope, up to
approximately 180 mm)

To compare the relative light grasp of two main lenses used at the
same magnification, compare the squares of their diameters.


FORMULAE FOR ASTROPHOTOGRAPHY

F-NUMBER: PRIME FOCUS (ERECT IMAGE)
f/ = F/D

where f/ is the f-number of the system (objective)
F is the focal length of the objective
D is the diameter of the objective


F-NUMBER: AFOCAL, EYEPIECE-CAMERA LENS (REVERSED IMAGE)
f/ = F'/D = (M*Fc)/D = ((F/Fe)*Fc)/D = (F/D)*(Fc/Fe) = (M/D)*Fc

where f/ is the f-number of the system
F' is the effective focal length of the system
Fe is the focal length of the ocular (divided by Barlow
magnification)
D is the diameter of the objective
M is the magnification
Fc is the focal length of the camera
F is the focal length of the objective
Fc/Fe is the projection magnification
M/D is the power per mm

The diameter of the first image equals the film diagonal
(44 mm for 35 mm film) divided by the magnification.


F-NUMBER: EYEPIECE PROJECTION, POSITIVE LENS (REVERSED IMAGE)
f/ = F'/D = (F/D)*(B/A) = (F/D)*(((M+1)*Fe)/A) = (F/D)*((B/Fe)-1)

where f/ is the f-number of the system
F' is the effective focal length of the system
D is the diameter of the objective
F is the focal length of the objective (times any Barlow
magnification)
B is the secondary image ("throw"), the distance of the
ocular center from the focal plane of the film,
equal to ((M+1)*Fe)/A
A is the primary image, the distance of the ocular center
from the focal point of the telescope objective
M is the projection magnification, equal to (B/Fe)-1
Fe is the focal length of the ocular


F-NUMBER: NEGATIVE LENS PROJECTION (ERECT IMAGE)
f/ = F'/D = (F/D) * (B/A)

where f/ is the f-number of the system
F' is the effective focal length of the system
D is the diameter of the objective
B is the distance of the Barlow center from the focal
plane of the film
A is the distance of the Barlow center from the focal
point of the telescope objective
B/A is the projection magnification (Barlow mag.)


EXPOSURE COMPARISON FOR EXTENDED OBJECTS
Exposure Compensation = (f/S)^2/(f/E)^2 = ((f/S)/(f/E))^2

(the ratio of intensities of illumination is squared
according to the inverse square law)

where Exposure Compensation is the exposure compensation
to be made to the example system
f/S is the f-number (f/) of the subject system
f/E is the f-number (f/) of the example system


EXPOSURE COMPARISON FOR POINT SOURCES
Exposure Compensation = De^2/Ds^2 = (De/Ds)^2

where Exposure Compensation is the exposure compensation
to be made to the example system
De is the objective diameter of the example system
Ds is the objective diameter of the subject system


LIGHT-RECORDING POWER OF A SYSTEM
Power = r^2/f^2

(the light-recording power is directly proportional to the
square of the radius of the objective and inversely propertional
to the square of the f-number)

where Power is the light-recording power of the system
r is the radius of the objective
f is the f-number (f/) of the system

Example: a 200-mm f/8 system compared with a 100-mm f/5 system
(100^2)/8^2 compared with (50^2)/5^2
156.25 compared with 100, or 1.56 times more
light-recording power


EFFICIENCY OF LENS FOR PHOTOGRAPHING AN AVERAGE METEOR
Efficiency = F/f^2

where Efficiency is the efficiency of the lens for
photographing an average meteor (in a meteor shower)
F is the focal length of the lens
f is the f-number (f/) of the lens


PRINT'S EFFECTIVE FOCAL LENGTH
Print EFL = Camera FL * Print Enlargement

where Print EFL is the print's effective focal length
Camera F. L. is the camera's focal length
Print Enlargement is the amount of enlargement
of the print (3x is the standard for 35-mm film)


GUIDESCOPE MAGNIFICATION
Guidescope M ~ f/12.5

where Guidescope M is the magnification needed
f is the photographic focal length in mm

Experience indicates that the minimum guiding magnification needed
is about f divided by 12.5, precisely what a 12.5 mm guiding
ocular used in an off- axis guider for prime-focus photography
yields. (Since visual magnification is the ratio of the objective
to ocular focal length, the combination of prime-focus camera and
off-axis guider with a 12.5-mm ocular gives a guiding
magnification of f/12.5. f/7.5 (as with a typical focal reducer
that reduces the effective focal length by a factor of 0.6) is a
significant improvement. f/5 or higher magnification is for
top-quality guiding.

Guidescope M = Guidescope EFL / Print EFL

where Guidescope M is the guidescope's magnification (should be >=
1, preferably 5-8)
Guidescope EFL is the guidescope's effective focal length, the
guidescope's focal length times any Barlow magnification (should
be >= to the focal length of the primary and the guidescope's
magnification, 0.2x per mm of focal length of the objective,
0.1x per mm of the camera lens
Print EFL is the print's effective focal length


GUIDING TOLERANCE
Guiding Tolerance = 0.076 * Guidescope M

where Guiding Tolerance is in mm
0.076 is one " at a 254-mm reading distance from the print
(a crosshair is usually 0.05 mm)


MAXIMUM ALLOWABLE TRACKING (SLOP) ERROR
S ~ 8250/(F*E)

where S is the error ("slop") in "
F is the focal length in mm
E is the amount of enlargement of the print
(3x is the standard for 35-mm film)

The slop is derived from the formula Theta = k*(h/F),
with k = 206256 (the number of seconds in a radian) and
h = 0.04 mm of image-drift tolerance (an empirical value from
astrophotographs).


CONVERSION OF PLATE SCALE TO EFFECTIVE FOCAL LENGTH
EFL = mm per degree * 57.3 = 206265/" per mm

where EFL is the effective focal length in mm
57.3 is the number of degrees in a radian
206256 is the number of " in a radian


RESOLVING POWER OF A PHOTOGRAPHIC SYSTEM
Resolving Power = 4191"/F

where Resolving Power is the resolving power of a photographic
system with Kodak 103a or color film
F is the focal length of the system in mm


MAXIMUM RESOLUTION FOR A PERFECT LENS
Maximum Resolution = 1600/f

where Maximum Resolution is the resolution for a perfect lens
f is the f-number (f/) of the lens

Most films, even fast ones, resolve only 60 lines/mm; the human
eye resolves 6 lines/mm (less gives a "wooly" appearance).
80 lines/mm for a 50-mm lens is rated excellent (equal to 1 min of
arc); a 200-mm lens is rated excellent with 40 lines/mm. 2415
films yields 320 line pairs (160 lines)/mm (equal to 1 second of
arc); Tri-X yields 80 lines/mm.


MINIMUM RESOLUTION NECESSARY FOR FILM
Minimum Resolution = Maximum Resolution * Print Enlargement

where Minimum Resolution is the min resolution necessary
Maximum Resolution is the max resolution for a perfect lens
Print Enlargement is the amount of enlargement of the print
(3x is the standard for 35-mm film)

 

SIZE OF IMAGE (ANGULAR)
h = (Theta*F)/k
Theta = k*(h/F)
F = (k*h)/Theta

where h is the linear height in mm of the image at prime focus
of an objective or a telephoto lens
Theta is the object's angular height (angle of view) in units
corresponding to k
F is the effective focal length (focal length times Barlow
magnification) in mm
k is a constant with a value of 57.3 for Theta in degrees, 3438 in
minutes of arc, 206265 for seconds of arc (the number of the
respective units in a radian)

The first formula yields image size of the sun and moon as
approximately 1% of the effective focal length
(Theta/k = 0.5/57.3 = 0.009).

The second formula can be used to find the angle of view (Theta)
for a given film frame size (h) and lens focal length (F).
Example: the 24 mm height, 36 mm width, and 43 mm diagonal of
35-mm film yields an angle of view of 27 deg, 41 deg, and 49 deg
for a 50-mm lens.

The third formula can be used to find the effective focal length
(F) required for a given film frame size (h) and angle of view
(Theta).

LENGTH OF A STAR TRAIL ON FILM
Length = F*T*0.0044

where Length is the length in mm of the star trail on film
F is the focal length of the lens in mm
T is the exposure time in minutes
0.0044 derives from (2*Pi)/N for minutes
(N = 1440 minutes per day)

EXPOSURE TIME FOR STAR TRAIL ON 35-MM FILM
T = 5455/F

where T is the exposure time in minutes for a length of 24 mm
(the smallest dimension of 35-mm film)
F is the focal length of the lens in mm


MAXIMUM EXPOSURE TIME WITHOUT STAR TRAIL
T = (1397/F)

where T is the maximum exposure time in seconds without a star
trail 1397 derives from 1' at reading distance (254 mm), the
smallest angular quantity that can be perceived by the human eye
without optical aid ("limiting resolution") and is equal to
< 0.1 mm. This quantity also applies to the moon. 2-3x yields
only a slight elongation. Use 20x for a clock drive.

F is the focal length of the lens in mm

The earth rotates 5' in 20 s, which yields a barely detectable
star trail with an unguided 50-mm lens. 2-3' (8-12 s) is
necessary for an undetectable trail, 1' (4 s) for an expert
exposure. Divide these values by the proportional increase in
focal length over a 50-mm lens. For example, for 3' (12 s), a
150-mm lens would be 1/3 (1' and 4 s) and a 1000-mm lens would be
1/20 (0.15' and 0.6 s). Note that to compensate for these values,
the constant in the formula would be 1000 for a barely-detectable
trail, 600 for an undetectable trail, and 200 for an expert
exposure.

N.B. The above formulae assume a declination of 0o. For other
declinations, multiply lengths and divide exposure times by the
following cosines of the respective declination angles: 0.98
(10o), 0.93 (20o), 0.86 (30o), 0.75 (40o), 0.64 (50o), 0.50 (60o),
0.34 (70o), 0.18 (80o), 0.10 (85o).


SIZE OF IMAGE (LINEAR)
i = (h/D)*F
h = (D*i)/F
D = (h*F)/i
F = (D*i)/h

where i is the linear image size in mm of the image at prime focus
of an objective or telephoto lens (for terrestrial objects, equal
to 24 mm divided by the amount of enlargement of the print [3x is
the standard for 35-mm film] for the smallest dimension of 35-mm
film])

h is the linear height of the object in units corresponding to D

D is the distance of the object in units corresponding to h

F is the effective focal length (focal length times Barlow
magnification) in mm

The last formula gives the focal length necessary to photograph a
recognizable celestial (Linear Width in km) or terrestrial (Linear
Width in m).


EXPOSURE DURATION FOR EXTENDED OBJECTS
E = f^2/(S*B)

where e is the exposure duration in seconds for an image size of
>= 0.1 mm

f is the f-number (f/) of the lens

S is the film's ISO speed

B is the brightness factor of the object (Venus 1000, Moon 125,
Mars 30, Jupiter 5.7)

Thus, a 2-minute exposure at f/1.4 is equivalent to a 32-minute
exposure at f/5.6 (4 stops squared times 2 minutes), ignoring the
effects of reciprocity failure in the film, which would mean that
the 32-minute exposure would have to be even longer.

SURFACE BRIGHTNESS OF AN EXTENDED OBJECT ("B" VALUE)
B = 10^0.4(9.5-M)/D^2

where B is the surface brightness of the (round) extended object

M is the magnitude of the object (total brightness of the object),
linearized in the formula

D is the angular diameter of the object in seconds of arc (D^2 is
the surface area of the object)


EXPOSURE DURATION FOR POINT SOURCES
e = (10^0.4(M+13))/S*a^2

where e is the exposure duration in seconds
for an image size of >= 0.1 mm
M is the magnitude of the object
S if the film's ISO speed
a is the aperture of the objective


MISCELLANEOUS FORMULAE

HOUR ANGLE
H = Theta - Delta

where H is the hour angle
Theta is sidereal time
Delta is right ascension

The Hour Angle is negative east of and positive west of the
meridian (as right ascension increases eastward).


BODE'S LAW
(4 + 3(2^n))/10 in AU at aphelion

where n is the serial order of the planets from the sun (Mercury's
2n =1, Venus's n = 0, Earth's n = 1, asteroid belt = 3)


APPARENT ANGULAR SIZE OF AN OBJECT
Theta = (h/D)*k

where Theta is the object's apparent angular size in units
corresponding to k

h is the linear height of the object in units corresponding to D

D is the distance of the object in units corresponding to h

Theta is the object's angular height (angle of view) in units
corresponding to k

k is a constant with a value of 57.3 for Theta in degrees, 3438 in
minutes of arc, 206265 for seconds of arc (the number of the
respective units in a radian)

A degree is the apparent size of an object whose distance is 57.3
times its diameter. The formula holds for celestial or
terrestrial objects. E.g., for the width of a quarter at arm's
length: (57.3*25 mm)/700 mm = 2o.

Under ideal conditions, the human eye can resolve anything
subtending more than a 1' angle, i.e., see an object as an
extended object or see a double star as two stars rather than a
single point of light, provided that the two components are of
nearly equal brightness. A more practical value would be 4'; 8'
is an even more practical value for comfortable viewing. The best
earthbound telescopes are usually limited by atmospheric effects
to objects 1" or larger (0.25" with excellent seeing) in apparent
size (before magnification). In theory, a telescope could see
everything with a magnification of 60x (1" magnified to 1').

LENGTH OF A METEOR TRAIL
h = (Theta*D)/57.3

where h is the linear height of the meteor in km
Theta is the object's apparent angular size in degrees
D is the distance of the object in km

GEOGRAPHIC DISTANCE
Geographic distance of one sec of arc = 30 m * cos of the latitude

where cos(Latitude)=1 on lines of constant longitude

ESTIMATING ANGULAR DISTANCE

Penny, 4 km distant ....................................... 1"
Sun, Moon ................................................. 30'
(The Moon is approximately 400 times smaller in angular
diameter than the Sun, but is approximately 400 times
closer.)
Width of little finger at arm's length .................... 1o
Dime at arm's length ...................................... 1o
Quarter at arm's length ................................... 2.5o
Width of Orion's belt ..................................... 3o
Alpha Ursae Majoris (Dubhe) to Beta Ursae Majoris (Merak) . 5o
(Height of Big Dipper's cup. These are the "pointer
stars" to Polaris.)
Alpha Geminorum (Castor) to Beta Geminorum (Pollux) ....... 5o
Width of fist at arm's length ............................. 10o
Alpha Ursae Majoris (Dubhe) to Delta Ursae Majoris (Megrez) 10o
(Width of Big Dipper's cup.)
Height of Orion ........................................... 16o
Length of palm at arm's length ............................ 18o
Width of thumb to little finger at arm's length ........... 20o
Alpha Ursae Majoris (Dubhe) to Eta Ursae Majoris (Alkaid) . 25o
(Length of Big Dipper.)
Alpha Ursae Majoris (Dubhe) to Alpha Ursae Minoris
(Polaris) ............................................. 27o


ESTIMATING MAGNITUDES

Big Dipper, from cup to handle
Alpha (Dubhe) 1.8
Beta (Merak) 2.4
Gamma (Phecda) 2.5
Delta (Megrez) 3.4
Epsilon (Alioth) 1.8
Zeta (Mizar) 2.2
Eta (Alkaid) 1.9

Little Dipper, from cup to handle
Beta (Kochab) 2.0
Gamma (Pherkad) 3.1
Eta 5.0
Zeta 4.3
Epsilon 4.4
Delta (Pherkard) 4.4
Alpha (Polaris) 2.0

RANGE OF USEFUL MAGNIFICATION OF A TELESCOPE

D = diameter of aperture in mm

Minimum useful magnification .................... 0.13*D 0.2*D for better contrast
Best visual acuity .............................. 0.25*D
Wide views ...................................... 0.4*D
Lowest power to see all detail (resolution of eye
matches resolution of telescope) ............. 0.5*D
Planets, Messier objects, general viewing ....... 0.8*D
Normal high power, double stars ................. 1.2*D to 1.6*D
Maximum useful magnification .................... 2.0*D
Close doubles ................................... 2.35*D
Sometimes useful for double stars ............... 4.0*D
Limit imposed by atmospheric turbulance ......... 500